Large time-stepping, delay-free, and invariant-set-preserving integrators for the viscous Cahn–Hilliard–Oono equation

Abstract

Stabilization techniques are frequently used to construct efficient and stable schemes for stiff and nonlinear systems. However, the introduced errors may lead to severe delay effects when facing large time-steps or stabilization parameters. Considering a fourth-order-in-space viscous Cahn–Hilliard–Oono equation that consists of both short- and long-range interactions, and equipped with either a polynomial or logarithmic potential, we analyze its invariant sets by proposing a novel reformulation. Additionally, we devise a family of temporal integrators up to eighth-order, along with a suitable stabilization parameter, to ensure the preservation of these invariant sets for any time-step. To eliminate the lagging phenomenon, we propose rewriting the method into a parametric two-step Runge–Kutta (TSRK) formulation, quantifying the rescaled time-step, and finally performing a relaxation technique. We demonstrate that the s-stage, pth-order parametric relaxation TSRK integrators retain the order of the underlying TSRK Butcher tableau, and promote the strong-stability-preserving coefficient to at most s when p>2. To further reduce the stabilization parameter, we propose a parametric relaxation integrating factor TSRK framework that treats the stiff linear operator as an integrating factor. The proposed framework has mild restrictions on the underlying TSRK tableau, i.e., the coefficients are non-negative and the parametric abscissas are non-decreasing. Numerical experiments demonstrate the accuracy and structure-preserving property of the proposed schemes, and are used to investigate the effect of parameters on microphase separations in diblock copolymer melts.